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Macroeconomics Exam Review 45

Macroeconomics Exam Review 45 - Solutions for Foundations...

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Conversely, assume that ( x 𝑛 ) is a sequence in 𝑛 with 𝑥 𝑛 𝑖 𝑥 𝑖 for every coordinate 𝑖 . Choose some 𝜖 > 0. For every 𝑖 , there exists some integer 𝑁 𝑖 such that 𝑥 𝑛 𝑖 𝑥 𝑖 < 𝜖 for every 𝑛 𝑁 𝑖 Let 𝑁 = max 𝑖 { 𝑁 1 , 𝑁 2 , . . . , 𝑁 𝑛 } . Then 𝑥 𝑛 𝑖 𝑥 𝑖 < 𝜖 for every 𝑛 𝑁 and x 𝑛 x = max 𝑖 𝑥 𝑛 𝑖 𝑥 𝑖 < 𝜖 for every 𝑛 𝑁 That is, x 𝑛 x . A similar proof can be given using the Euclidean norm ∥⋅∥ 2 , but it is slightly more complicated. This illustrates an instance where the sup norm is more tractable. 1.215 1. Let 𝑆 𝑋 be closed and bounded and let x 𝑚 be a sequence in 𝑆 . Every term x 𝑚 has a representation x 𝑚 = 𝑛 𝑖 =1 𝛼 𝑚 𝑖 x 𝑖 Since 𝑆 is bounded, so is x 𝑚 . That is, there exists 𝑘 such that x 𝑚 ∥ ≤ 𝑘 for all 𝑚 . Applying Lemma 1.1, there is a positive constant 𝑐
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